Kohn--Nirenberg quantization of the affine group and related examples

We show how to construct unitary dual $2$-cocycles for a class of semidirect products that exhibit many similarities with the affine group ${\rm Aff}(V)=\GL(V)\ltimes V$ of a finite dimensional vector space over a local skew field. The primary source of examples comes from Lie groups whose Lie algebras are Frobenius seaweeds. The construction builds on our earlier results and relies heavily on representation theory and an associated quantization procedure of Kohn--Nirenberg type. On the technical side, the key point is the observation that any semidirect product $G=H\ltimes V$ in our class can be presented as a double crossed product $G=P\bowtie N$ with respect to which the unique square-integrable irreducible representation of $G$ takes a particularly nice form. The Kohn--Nirenberg quantization that we construct is intimately related to a scalar Fourier transform $\CF\colon L^2(N)\to L^2(P)$ intertwining the left regular representations of $P$ and $N$ with representations defined by the dressing transformations.